Stability Rates for Linear Ill-Posed Problems with Convolution and Multiplication Operators

In this paper we deal with thèstrength' of ill-posedness for ill-posed linear operator equations Ax = y in Hilbert spaces, where we distinguish according to M. Z. Nashed 15] the ill-posedness of type I if A is not compact, but we have R(A) 6 = R(A) for the range R(A) of A; and the ill-posedness of type II for compact operators A: From our considerations it seems to follow that the problems with noncompact operators A are not in generaìless' ill-posed than the problems with compact operators. We motivate this statement by comparing the approximation and stability behaviour of discrete least-squares solutions and the growth rate of Galerkin matrices in both cases. Ill-posedness measures for compact operators A as discussed in 10] are derived from the decay rate of the nonincreasing sequence of singular values of A. Since singular values do not exist for noncompact operators A; we introduce stability rates in order to have a common measure for the compact and noncompact cases. Properties of these rates are illustrated by means of convolution equations in the compact case and by means of equations with multiplication operators in the noncompact case. Moreover using increasing rearrangements of the multiplier functions speciic measures of ill-posedness called ill-posedness rates are considered for the multiplication operators. In this context, the character of suucient conditions providing convergence rates of Tikhonov regularization are compared for compact operators and multiplication operators.